View source for Category Theory/(Co-)cones and (co-)limits

{{definition|cone|Let <math>\mathcal C</math> be a category and let <math>G = (V, A)</math> be a diagram in <math>\mathcal C</math>. A '''cone''' over <math>G</math> is an object <math>a</math> of <math>\mathcal C</math>, together with morphisms <math>\phi_{a, b}: a \to b</math> for each <math>b \in V</math>, so that for each <math>\psi: b \to b' \in A</math> (so that <math>b, b' \in V</math>) we have
:<math>\phi_{a, b'} =  \psi \circ \phi_{a, b}</math>.}}

{{definition|limit|Let <math>\mathcal C</math> be a category, let <math>G = (V, A)</math> be a diagram in <math>\mathcal C</math> and let }}

{{definition|diagonal|Let <math>\mathcal C</math> be a category, and let <math>X</math> be an object of <math>\mathcal C</math>. Further, suppose that the object <math>X \times X</math> and the morphisms <math>\pi_1: X \times X \to X</math> and <math>\pi_2: X \times X \to X</math> constitute a product of <math>X</math> with itself in <math>\mathcal C</math>. Then the '''diagonal''' (also called "diagonal morphism") is the unique morphism
:<math>\Delta: X \to X \times X</math>
such that <math>\pi_1 \circ \Delta = \pi_2 \circ \Delta = \operatorname{Id}_X</math>.}}

{{definition|anti-diagonal|Let <math>\mathcal C</math> be a category, and let <math>X</math> be an object of <math>\mathcal C</math>. Further, suppose that the object <math>X \sqcup X</math> and the morphisms <math>\iota_1: X \to X \sqcup X</math> and <math>\iota_2: X \to X \sqcup X</math> constitute a coproduct of <math>X</math> with itself in <math>\mathcal C</math>. Then the '''anti-diagonal''' (also called "anti-diagonal morphism") is the unique morphism
:<math>\nabla: X \sqcup X \to X</math>
such that <math>\nabla \circ \iota_1 = \nabla \circ \iota_2 = \operatorname{Id}_X</math>.}}

== Exercises ==
#
##Let <math>\mathcal D</math> be a category such that every two objects of <math>\mathcal D</math> have a product. Suppose further that <math>\mathcal C</math> is another category, and that <math>T: \mathcal C \to \mathcal D</math> is a functor. Let <math>Y</math> be an object of <math>\mathcal D</math>. Use the universal property of the product in order to show that there exists a functor <math>S_Y: \mathcal C \to \mathcal D</math> that sends an object <math>X</math> of <math>\mathcal C</math> to the object <math>T(X) \times Y</math> of <math>\mathcal D</math>.
##Prove that any morphism <math>g: Y \to Z</math> in <math>\mathcal D</math> gives rise to a natural transformation <math>S_Y \to S_Z</math>.
##Can we weaken the assumption that ''every'' two objects of <math>\mathcal D</math> have a product? (Hint: Consider the image of the class function on objects associated to the functor <math>T</math>.)

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